Closely related to the change point testing literature,
thresholding is popularly adopted for model selection in change point detection.
[1]} derive the (distributional) limit behaviour of the MOSUM test statistic
\(\max _{k \in \mathcal {S}_{\text{M}}(G)} \vert \mathcal {T}_{s, k, e}(X) \vert \)  under the null hypothesis of no change,
imposing general conditions allowing for heavy-tailedness and serial dependence in \(\lbrace \varepsilon _t\rbrace \) 
with the bandwidth \(G\)  satisfying \(G \rightarrow \infty \)  while \(G/n \rightarrow 0\) ;
the result is extended for the asymmetric MOSUM statistic in [2]}.
[3]}, [4]} approximate the MOSUM statistic
by a Gaussian process when \(G\)  grows linearly with \(n\) .
With such results, we can test \(H_0: \, q_n = 0\)  against \(H_1: \, q_n > 0\) 
with a theoretically motivated threshold at a chosen significance level.
